Statistical Ensembles of Complex, Quaternion, and Real Matrices

Statistical ensembles of complex, quaternion, and real matrices with Gaussian probability distribution, are studied. We determine the over-all eigenvalue distribution in these three cases (in the real case, under the restriction that all eigenvalues are real). We also determine, in the complex case, all the correlation functions of the eigenvalues, as well as their limits when the order N of the matrices becomes infinite. In particular, the limit of the eigenvalue density as N → ∞ is constant over the whole complex plane.